The value of $(1 + tan 1^{o}) (1 + tan 2^{o}) (1 + tan 3^{o}) . . . . . . . . . . . . (1 + tan 45^{o})$ is equal to

2^{21}

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2^{22}

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2^{23}

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2^{24}

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Solution

The correct option is C $2^{23}$
As we know that,
$(1 + t a n x) (1 + t a n (45 - x)) = (1 + t a n) (1 + \frac{t a n 45 - t a n x}{1 + t a n 45 t a n x})$

$= (1 + t a n x) (\frac{1 + t a n x + 1 - t a n x}{1 + t a n x})$

$= 2$

$⟹ (1 + t a n 1) (1 + t a n 2) (1 + t a n 3) . . . . . . . . . . . . (1 + t a n 45)$

$= (1 + t a n 1) (1 + t a n 44) (1 + t a n 2) (1 + t a n 43) . . . . . . (1 + t a n 45)$

$= (1 + t a n 1) (1 + t a n (45 - 1)) (1 + t a n 2) (1 + t a n (45 - 2)) . . . . . . . (1 + 1)$

$= 2 \times 2 \times . . . .$ upto 22 times $\times 2$

$= 2^{22} \times 2 = 2^{23}$

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